MatheMatics a Practical Odyssey
David B. Johnson
Diablo Valley College
Pleasant Hill, California
Thomas A. Mowry
Diablo Valley College
Pleasant Hill, California
8e
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Mathematics: A Practical Odyssey, Eighth Edition
David Johnson, Thomas Mowry
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Overview ix
chapter 1 Logic 1
1.1 Deductive versus Inductive Reasoning 2
1.2 Symbolic Logic 18
1.3 Truth Tables 29
1.4 More on Conditionals 41
1.5 Analyzing Arguments 49
1.6 Deductive Proof of Validity 61
chapter
2 Sets and counting 73
2.1 Sets and Set Operations 74
2.2 Applications of Venn Diagrams 86
2.3 Introduction to Combinatorics 99
2.4 Permutations and Combinations 106
2.5 Infinite Sets 122
chapter
3 probability 137
3.1 History of Probability 138
3.2 Basic Terms of Probability 145
3.3 Basic Rules of Probability 164
3.4 Combinatorics and Probability 177
3.5 Expected Value 190
3.6 Conditional Probability 20 1
3.7 Independence, Medical Tests, and Genetics 216
chapter 4
Statistics 233
4.1 Population, Sample, and Data 234
4.2 Measures of Central Tendency 258
4.3 Measures of Dispersion 271
Contents
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chApTer
chApTer
chApTer
chApTer
4.4 The Normal Distribution 288
4.5 Polls and Margin of Error 306
4.6 Linear Regression 320
5 Finance 341
5.1 Simple Interest 342
5.2 Compound Interest 353
5.3 Annuities 367
5.4 Amortized Loans 379
5.5 Annual Percentage Rate with a TI’s TVM Application 400
5.6 Payout Annuities 408
6
Voting and Apportionment 421
6.1 Voting Systems 422
6.2 Methods of Apportionment 442
6.3 Flaws of Apportionment 470
7
Number Systems and Number Theory 485
7.1 Place Systems 486
7.2 Addition and Subtraction in Different Bases 501
7.3 Multiplication and Division in Different Bases 506
7.4 Prime Numbers and Perfect Numbers 511
7.5 Fibonacci Numbers and the Golden Ratio 523
8
Geometry 537
8.1 Perimeter and Area 538
8.2 Volume and Surface Area 556
8.3 Egyptian Geometry 568
8.4 The Greeks 578
8.5 Right Triangle Trigonometry 590
8.6 Linear Perspective 606
8.7 Conic Sections and Analytic Geometry 615
vi CONTENTS Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
8.8 Non-Euclidean Geometry 627
8.9 Fractal Geometry 636
8.10 The Perimeter and Area of a Fractal 653
chapter 9 Graph theory 669
9.1 A Walk through Königsberg 670
9.2 Graphs and Euler Trails 676
9.3 Hamilton Circuits 688
9.4 Networks 701
9.5 Scheduling 716
chapter 10 exponential and Logarithmic Functions 735
10.0A Review of Exponentials and Logarithms 736
10.0B Review of Properties of Logarithms 746
10.1 Exponential Growth 759
10.2 Exponential Decay 776
10.3 Logarithmic Scales 793
chapter 11
Markov chains 811
11.0A Review of Matrices 81 2
11.0B Review of Systems of Linear Equations 824
11.1 Markov Chains and Tree Diagrams 831
11.2 Markov Chains and Matrices 835
11.3 Long-Range Predictions with Markov Chains 843
11.4 Solving Larger Systems of Equations 848
11.5 More on Markov Chains 853
chapter
12 Linear programming 861
12.0 Review of Linear Inequalities 862
12.1 The Geometry of Linear Programming 874
12.2 Introduction to the Simplex Method 12-2
12.3 The Simplex Method: Complete Problems 12-8
CONTENTS vii
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Overview
The goal of Mathematics: A Practical Odyssey is to expose students to the utility, relevance, and beauty of mathematics in the context of every-day themes and across multiple disciplines, and to broaden the narrow view of mathematics that may come from an isolated study of algebra. The text incorporates many items of interest, including historical notes, articles, and discussions of contemporary issues along with many rich illustrations and fine art to demonstrate a wide range of topics. We believe that students who engage with the content in this text will have a broader outlook and understanding of the world around them, in the spirit of a true liberal arts education. They will benefit not only from the analytical tools and mathematical skills they practice and acquire, but from the references to important scientific research and discoveries, as well as works of literature, history, art, and politics.
The following list is meant to make the text’s goals more concrete. Skim the list to learn more about typical outcomes for each chapter. Also use the list to identify areas that you might cover in your course.
l In Chapter 1, Logic, learn to analyze the validity of an argument.
l In Chapter 2, Sets and Counting, and Chapter 3, Probability, learn to understand the risks of inherited diseases and the outcomes associated with lotteries and bets.
l In Chapter 4, Statistics, learn to understand the accuracy and validity of a public opinion poll.
l In Chapter 5, Finance, learn the ins and outs of buying a house or car, and using student loans to finance your college education.
l In Chapter 6, Voting and Apportionment, learn that there is no perfect voting system or method of apportionment.
l In Chapter 7, Number Systems and Number Theory, learn about the origins of commonly used number systems and their applications and learn about the expression of the Fibonacci numbers in nature and the golden ratio art.
l In Chapter 8, Geometry, learn to understand its origins and applications.
l In Chapter 9, Graph Theory, learn to create networks and apply graph theory to schedules.
l In Chapter 10, Exponential and Logarithmic Functions, learn how populations grow, how radiocarbon dating works, how the Richter scale measures earthquakes, and how sound is measured in decibels.
l In Chapter 11, Markov Chains, learn how manufacturers can predict their products’ success or failure in the marketplace.
l In Chapter 12, Linear Programming, learn how a small business can determine how to utilize limited resources to maximize its profit.
l In Chapter 13, Calculus, learn more about the subject and its uses. (Note: this chapter does not appear within the textbook, but via www.cengagebrain. com only.)
ix Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
course Level and Flexibility
Mathematics: A Practical Odyssey is written for the student who has a working knowledge of intermediate algebra, not the student who has a perfect mastery of it. The student is expected to use this knowledge to acquire critical thinking and quantitative reasoning skills. Even though some chapters are not algebra-based, it could be challenging for a student without background in intermediate algebra to succeed in a course using this book. While the specific skills students learn in intermediate algebra courses do not always directly apply, the chapters require a level of critical thinking and mathematical maturity more commonly found in students who have passed intermediate algebra.
This book contains a wide range of topics. As much as possible, it has been written so that its chapters are independent of each other. Instructors have wide latitude in selecting topics to cover and can therefore customize the course so that it is responsive to the needs of each instructor’s individual classes and institution. In some cases, a chapter has a suggested core of key sections as well as a selection of optional sections that offer enrichment. These sections are not labeled as “core” and “optional” in the book; rather, this distinction is made in the Chapter Summaries that are found in the Instructor’s Resource Manual (which can be accessed via the Instructor’s Companion Site at www.cengagebrain.com).
While instructors may pick and choose chapters to suit the needs of the course, we do recommend that a few topics precede others. Please see the Instructor’s Resource Manual posted on the Instructor Companion Site for more specifics on content connections and dependencies as outlined in the Chapter Summaries.
New to the eighth edition
In Chapter 1, Logic:
l a new section 1.6 on Deductive Proof of Validity appears. In this section, elementary valid argument forms and common rules of replacement are developed; formal proofs of validity are then constructed by applying the basic rules of inference.
l the four standard form categorical proposition types are now included: the universal affirmative, the universal negative, the particular affirmative, and the particular negative.
l all conditional statements of the form “p implies q” now use standard terminology in which p is referred to as the “antecedent” and q the “consequent.”
In Chapter 3, Probability:
l hemophilia is now covered along with other inherited diseases.
l the treatment of Simpson’s paradox has been expanded.
In Chapter 4, Statistics:
l the distinction between sample and population standard deviation is introduced and explored in new exercises.
x OVERVIEW
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In Chapter 5, Finance:
l there is a new focus on student finance, including student loans and maxed-out credit cards.
In Chapter 7, Number Systems and Number Theory:
l there is a new emphasis on how certain bases, including base four, base five, base eight, base ten, base twelve, and base twenty, correspond to humanity’s historical use of different body parts in counting, and why base sixty is used to keep time and to measure angles.
Chapter 11, Markov Chains, has been rewritten so that the material is more accessible, and so that review topics more closely fit our just-enough, just-intime review policy.
In many chapters, especially Chapter 3 on Probability, and Chapter 5 on Finance, discussions and examples were heavily streamlined and given more structure to support students.
Exercises that require that the student engage in hands-on activities are now included. These same exercises can be used for in-class activities.
Throughout the textbook, real-world data within examples and exercises have been updated.
“Rough checks” now teach the student how to determine if an answer is reasonable; that is, if it’s “in the ballpark.”
OVERVIEW xi
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chapter Openers
The chapter openers provide background and motivation for the study of the topics within each chapter. “What We Will Do in This Chapter,” gives a preview of concepts and topics that will be explored.
s Features examples
sThe worked-out examples contain detailed steps that effectively demonstrate solution methods. There is a special emphasis placed on checking that solutions are reasonable and presented in a form that is mathematically appropriate. Some of the checks are specially called out as “Rough Checks” to help students get in the habit of determining whether an answer makes sense. Other “Calculation Notes” provide tips to help students avoid common errors. Both types of checks can be found following the solutions of selected examples.
xii OVERVIEW
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sTopic X
Topic X provides an opportunity to reinforce the practical emphasis of the text by illustrating current, powerful, realword uses of the text’s topics. The exercises that relate back to Topic X, included among the end-of-section exercises, take on increased significance and meaning as students see the text concepts put into direct context.
Featured in the News Newspaper and magazine articles illustrate how the book’s topics come up in the real world. s
Found throughout the text, historical Notes highlight the work of important people or events that relate to topics or concepts under discussion. s
OVERVIEW xiii
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exercises
The section-ending exercises are fresh, relevant, and span a wide variety of types. They are designed to solidify students’ understanding of the material and make them proficient in the calculations involved.
In sections where Topic X appears, exercises relating back to the featured discussion are included.
The short-answer history questions are meant to focus and reinforce the students’ understanding of the historical material.
concept questions test students understanding of ideas, often asking them to provide their own examples or explain main ideas in their own words.
The hands On exercises and projects provide opportunites for further exploration or enrichment, often requiring students to try something first-hand or perform more open-ended research.
sThe Next Level
These special exercises are designed to help prepare students for admission examinations such as the GRE (required for graduate school) or the GMAT (required for graduate study in business).
Answers to the odd-numbered exercises are given in the back of the book, with two exceptions:
l Answers to historical questions and essay questions are not given.
l Answers are not given when the exercises instruct the student to check the answers themselves.
xiv OVERVIEW
s
s
s
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s
GrAphING AND ScIeNTIFIc cALcuLATOrS
Calculator boxes provide all of the necessary keystrokes for both scientific calculators and graphing calculators. Calculator subsections go beyond the keystrokes to offer more in-depth guidance.
The calculator boxes often offer keystrokes for several models of calculator, to provide optimal support.
OVERVIEW xv
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sMicrosoft excel©
A number of subsections that give instruction on the use of Excel are included. The following topics are addressed:
l Section 4.1: Histograms and Pie Charts on a Computerized Spreadsheet (page 251)
l Section 4.3: Measures of Central Tendency and Dispersion on Excel (page 285)
l Section 5.4: Amortized Loans (page 396)
These subsections allow instructors to incorporate the computer into their class if they so desire, but they are entirely optional, and the book is in no way computer dependent. The subsections do not assume any previous experience with Excel.
Algebra review
Algebra reviews are justenough, just-in-time. Algebra topics that may not have been covered in the students’ algebra classes and topics to which students typically need multiple exposures are reviewed in sections placed immediately prior to the location where the topics are used, usually in a “section zero” at the beginning of the chapter. Those review sections cover only what is needed. More basic topics such as equation solving are not formally reviewed. Rather, the examples are selected so that they both explain the new material and review the appropriate algebra at the same time.
xvi OVERVIEW
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s
chapter review and review exercises
At the end of every chapter is a Review including important Terms, Formulas, and summary of procedures or steps for solving problems found in the chapter. Following the Review are exercises similar to those found in each section of the chapter for students to use in preparation for a test or exam.
OVERVIEW xvii
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s
TO The STuDeNT, AS yOu eMBArk ON yOur ODySSey . . .
This textbook is designed for students in liberal arts programs and other fields that do not require a core of mathematics. The term liberal arts is a translation of a Latin phrase that means “studies befitting a free person.” It was applied during the Middle Ages to seven branches of learning: arithmetic, geometry, logic, grammar, rhetoric, astronomy, and music. You might be surprised to learn that almost half of the original liberal arts are mathematics subjects.
In accordance with the tradition, handed down from the Middle Ages, that a broad-based education includes some mathematics, many institutions of higher education require their students to complete a college-level mathematics course. These schools award a bachelor’s degree to a person who not only has acquired a detailed knowledge of his or her field but also has a broad background in the liberal arts.
The goal of this textbook is to expose you to topics in mathematics that are usable and relevant to any educated person. We hope that you will encounter topics that will be useful at some time during your life. In addition, you are encouraged to recognize the relevance of mathematics to a well-rounded education and to appreciate the creative, human aspect of mathematics.
Your algebra background doesn’t have to be perfect, but algebra will come up. It’s also true that this book is written for a college-level math course. For the best result, you will have to work hard and put in a solid effort.
Your success in this course is important to us. To help you achieve that success, we have incorporated features in the textbook that promote learning and support various learning styles. Among these features are algebra review and instructions in using a calculator.
Our algebra reviews occur at the very beginning of the chapters, and they review only the algebra that comes up in that chapter. There are many topics in algebra in which students need to review and sharpen their skills. Calculator boxes give you all of the necessary keystrokes for scientific calculators and for graphing calculators. Calculator subsections help you learn how to use your calculator when a list of keystrokes is just not enough. We encourage you to examine these features and use them on your successful odyssey throughout this course.
xviii OVERVIEW
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SuppLeMeNTS
FOr The STuDeNT
Student Solutions Manual (ISBN: 978-1-305-10863-9)
The Student Solutions Manual provides worked-out solutions to odd-numbered problems in the text. Use of the solutions manual ensures that students learn the correct steps to arrive at an answer. (Print)
Text Specific DVDs (ISBN: 1111571570)
Hosted by Dana Mosely, these professionally produced DVDs cover key topics of the text, offering a valuable alternative for classroom instruction or independent study and review. (Media)
enhanced WebAssign®
Instant Access Code: 978-1-285-85803-6
Printed Access Card: 978-1-285-85802-9
Enhanced WebAssign combines exceptional mathematics content with the most powerful online homework solution, WebAssign. Enhanced WebAssign engages students with immediate feedback, rich tutorial content, and an interactive, fully customizable eBook, the Cengage YouBook, helping students to develop a deeper conceptual understanding of their subject matter.
cengageBrain.com
To access additional course materials, please visit www. cengagebrain.com. At the CengageBrain.com home page, search for the ISBN (from the back cover of your book) of your title using the search box at the top of the page. This will take you to the product page where these resources can be found.
FOr The INSTrucTOr
Instructor’s edition (ISBN: 978-1-305-10418-1)
The Instructor’s Edition features an appendix containing the answers to all problems. (Print)
Instructor’s resource Manual (ISBN: 978-1-305-11309-1)
The Instructor’s Resource Manual provides worked-out solutions to all of the problems in the text and includes suggestions for course syllabi and chapter summaries. This manual can be found on the Instructor Companion Site.
Text Specific DVDs (ISBN: 1111571570)
Hosted by Dana Mosely, these professionally produced DVDs cover key topics of the text, offering a valuable alternative for classroom instruction or independent study and review. (Media)
enhanced WebAssign®
Instant Access Code: 978-1-285-85803-6
Printed Access Card: 978-1-285-85802-9
Enhanced WebAssign combines exceptional mathematics content with the most powerful online homework solution, WebAssign. Enhanced WebAssign engages students with immediate feedback, rich tutorial content, and an interactive, fully customizable eBook, the Cengage YouBook, helping students to develop a deeper conceptual understanding of their subject matter. Visit www.cengage.com/ewa to learn more.
Instructor companion Site
Everything you need for your course in one place! This collection of book-specific lecture and class tools is available online via www.cengage.com/login. Access and download PowerPoint® presentations, images, solutions manual, and more.
cengage Learning Testing powered by cognero®
Instant Access Code: 978-1-305-26638-4
Cognero is a flexible, online system that allows you to author, edit, and manage test bank content, create multiple test versions in an instant, and deliver tests from your LMS, your classroom or wherever you want. This is available online via www.cengage.com/login.
OVERVIEW xix
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Acknowledgments
The authors would like to thank Richard Stratton, Jennifer Cordoba, Samantha Lugtu, Tanya Nigh, Vernon Boes, Erin Brown, and all the fine people at Cengage Learning. We also thank Scott Barnett, Jon Booze, and Kristy Hill for their efforts with accuracy and solutions manual authoring.
Special thanks go to the users of the text and reviewers who evaluated the manuscript for this edition, as well as those who offered comments on previous editions.
reviewers
Dennis Airey, Rancho Santiago College
Francisco E. Alarcon, Indiana University of Pennsylvania
Judith Arms, University of Washington
Bruce Atkinson, Palm Beach Atlantic College
Wayne C. Bell, Murray State University
Wayne Bishop, California State University—Los Angeles
David Boliver, Trenton State College
Stephen Brick, University of South Alabama
Barry Bronson, Western Kentucky University
Frank Burk, California State University—Chico
Laura Cameron, University of New Mexico
Jack Carter, California State University—Hayward
Timothy D. Cavanaugh, University of Northern Colorado
Joseph Chavez, California State University—San Bernadino
Eric Clarkson, Murray State University
Rebecca Conti, State University of New York at Fredonia
S.G. Crossley, University of Southern Alabama
Ben Divers, Jr., Ferrum College
Al Dixon, College of the Ozarks
Joe S. Evans, Middle Tennessee State University
Hajrudin Fejzie, California State University—San Bernardino
Lloyd Gavin, California State University— Sacramento
William Greiner, McLennan Community College
Martin Haines, Olympic College
Ray Hamlett, East Central University
Virginia Hanks, Western Kentucky University
Brian Heaven, Tacoma Community College
Anne Herbst, Santa Rosa Junior College
Linda Hinzman, Pasadena City College
Thomas Hull, University of Rhode Island
Robert W. Hunt, Humboldt State University
Robert Jajcay, Indiana State University
Irja Kalantari, Western Illinois University
Daniel Katz, University of Kansas
Palmer Kocher, SUNY, New Paltz
Katalin Kolossa, Arizona State University
Donnald H. Lander, Brevard College
Lee LaRue, Paris Junior College
Mike LeVan, Transylvania University
Lowell Lynde, University of Arkansas at Monticello
Thomas McCready, California State University— Chico
Vicki McMillian, Stockton State University
Narendra L. Maria, California State University— Stanislaus
John Martin, Santa Rosa Junior College
Gael Mericle, Mankato State University
Robert Morgan, Pima Community College
Pamela G. Nelson, Panhandle State University
Carol Oelkers, Fullerton College
Michael Olinick, Middlebury College
Matthew Pickard, University of Puget Sound
Joan D. Putnam, University of Northern Colorado
J. Doug Richey, Northeast Texas Community College
Stewart Robinson, Cleveland State University
Catherine Sausville, George Mason University
Eugene P. Schlereth, University of Tennessee at Chattanooga
Gary Shufelt, Muhlenberg College
Lawrence Somer, Catholic University of America
Charles Stevens, Skagit Valley Technical College
Laurence Stone, Dakota County Technical College
Charles Ziegenfus, James Madison University
Michael Trapuzzano, Arizona State University
Pat Velicky, Mid-Plains Community College
Karen M. Walters, University of Northern California
Dennis W. Watson, Clark College
Denielle Williams, Eastern Washington University
Charles Ziegenfus, James Madison University
xx OVERVIEW
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When writer Lewis Carroll took Alice on her journeys down the rabbit hole to Wonderland and through the looking glass, she had many fantastic encounters with the tea-sipping Mad Hatter, a hookah-smoking Caterpillar, the Cheshire Cat, and Tweedledee and Tweedledum. On the surface, Carroll’s writings seem to be delightful nonsense and mere children’s entertainment. Many people are quite surprised to learn that Alice’s Adventures in Wonderland is as much an exercise in logic as it is a fantasy and that Lewis Carroll was actually Charles Dodgson, an Oxford mathematician. In addition to “Alice,” Dodgson’s many writings include the whimsical The Game of Logic and the brilliant Symbolic Logic. Logic has fascinated scholars, philosophers, detectives, and star ship officers from the ancient, classic Greeks, to the ecentric, violin-playing Sherlock Holmes, to the deadpan, emotionless Mr. Spock. In today’s world of misleading commercial claims, innuendo, and political rhetoric, it is imperative that we employ logic to distinguish valid from invalid arguments; consequently, armed with the fundamentals of logic, we can “live long and prosper.”
What We Will Do In This Chapter
l We will study the basic components of logic and its application.
l We will explore different types of reasoning (deductive and inductive) and how they are applied to puzzles, critical thinking, and problem solving.
(continued)
© iStockPhoto.com/Bart
1 1
Sadowski
logic
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What We Will Do In This Chapter (continued)
l We will analyze and explore various types of statements and the conditions under which they are true. This process will be steamlined by translating verbal statements into symbolic form.
l We will explore equivalent forms of statements, including those of the conditional form “if then . . .”. In so doing, we will learn the difference between a “necessary condition” and a “sufficient condition.”
l We will use various methods including Venn diagrams, truth tables, and formal deductive proofs to determine the validity of an argument.
l We will look at the lives and accomplishments of some of the influential people who have shaped the study logic.
1.1
DeDuctive versus inDuctive reasoning
objectives
l Use Venn diagrams to determine the validity of deductive arguments
l Use inductive reasoning to predict patterns
Logic is the science of correct reasoning. Webster’s New World College Dictionary defines reasoning as “the drawing of inferences or conclusions from known or assumed facts.” Reasoning is an integral part of our daily lives; we take appropriate actions based on our perceptions and experiences. For instance, if you always encounter a traffic jam when taking a specific route while driving to school, you may decide to take an alternate route or leave home earlier on the day of an exam!
2
Logic is the science of correct reasoning. Auguste Rodin captured this ideal in his bronze sculpture The Thinker.
Vanni Archive/Art Resource, NY
In their quest for logical perfection, the Vulcans of Star Trek abandoned all emotion. Mr. Spock’s frequent proclamation that “emotions are illogical” typified this attitude.
Paramount
Television/The Kobal Collection/Picture Desk
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n Problem solving
Logic and reasoning are associated with the phrases problem solving and critical thinking. If we are faced with a problem, puzzle, or dilemma, we attempt to reason through it in hopes of arriving at a solution.
The first step in solving any problem is to define the problem in a thorough and accurate manner. Although this might sound like an obvious step, it is often overlooked. Always ask yourself, “What am I being asked to do?” Before you can solve a problem, you must understand the question. Once the problem has been defined, all known information that is relevant to it must be gathered, organized, and analyzed. This analysis should include a comparison of the present problem to previous ones. How is it similar? How is it different? Does a previous method of solution apply? If it seems appropriate, draw a picture of the problem; visual representations often provide insight into the interpretation of clues.
Before using any specific formula or method of solution, determine whether its use is valid for the situation at hand. A common error is to use a formula or method of solution when it does not apply. If a past formula or method of solution is appropriate, use it; if not, explore standard options and develop creative alternatives. Do not be afraid to try something different or out of the ordinary. “What if I try this . . . ?” may lead to a unique solution.
n
Deductive reasoning
Once a problem has been defined and analyzed, it might fall into a known category of problems, so a common method of solution may be applied. For instance, when one is asked to solve the equation x2 5 2x 1 1, realizing that it is a second-degree equation (that is, a quadratic equation) leads one to put it into the standard form (x2 2x 1 5 0) and apply the Quadratic Formula.
examPle 1
Using dedUctive reasoning to soLve an eqUation Solve the equation x2 5 2x 1 1.
The given equation is a second-degree equation in one variable. We know that all second-degree equations in one variable (in the form ax2 1 bx 1 c 5 0) can be solved by applying the Quadratic Formula:
1.1 Deductive versus Inductive Reasoning 3
Using his extraordinary powers of logical deduction, Sherlock Holmes solves another mystery. “Finding the villain was elementary, my dear Watson.”
Avco Embassy/The Kobal Collection/Picture Desk
x 5 b 6 Ïb2 4ac 2a
x2 5
x 1
x2 5 2x 1 1 x2 2x 1 5 0 x 5 s 2d 6 Ïs 2d2 s4ds1ds 1d 2s1d x 5 2 6 Ï4 1 4 2 SoluTIon Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Therefore,
2
1 can be solved by applying the Quadratic Formula:
The solutions are x 5 1 1 Ï2 and x 5 1 Ï2
In Example 1, we applied a general rule to a specific case; we reasoned that it was valid to apply the (general) Quadratic Formula to the (specific) equation x2 5 2x 1 1. This type of logic is known as deductive reasoning—that is, the application of a general statement to a specific instance.
Deductive reasoning and the formal structure of logic have been studied for thousands of years. One of the earliest logicians, and one of the most renowned, was Aristotle (384–322 b c ). He was the student of the great philosopher Plato and the tutor of Alexander the Great, the conqueror of all the land from Greece to India. Aristotle’s philosophy is pervasive; it influenced Roman Catholic theology through St. Thomas Aquinas and continues to influence modern philosophy. For centuries, Aristotelian logic was part of the education of lawyers and politicians and was used to distinguish valid arguments from invalid ones.
For Aristotle, logic was the necessary tool for any inquiry, and the syllogism was the sequence followed by all logical thought. A syllogism is an argument composed of two statements, or premises (the major and minor premises), followed by a conclusion. For any given set of premises, if the conclusion of an argument is guaranteed (that is, if it is inescapable in all instances), the argument is valid. If the conclusion is not guaranteed (that is, if there is at least one instance in which it does not follow), the argument is invalid.
Perhaps the best known of Aristotle’s syllogisms is the following:
1. All men are mortal. major premise
2. Socrates is a man. minor premise
Therefore, Socrates is mortal. conclusion
When the major premise is applied to the minor premise, the conclusion is inescapable; the argument is valid.
Notice that the deductive reasoning used in the analysis of Example 1 has exactly the same structure as Aristotle’s syllogism concerning Socrates:
1. All second-degree equations in one variable can be major premise solved by applying the Quadratic Formula.
2. x2 5 2x 1 1 is a second-degree equation in one variable. minor premise
Therefore, x2 5 2x 1 1 can be solved by applying the conclusion Quadratic Formula.
Each of these syllogisms is of the following general form:
1. If A, then B. all A are B. (major premise)
2. x is A. We have A. (minor premise)
Therefore, x is B. therefore, we have B. (conclusion)
Historically, this valid pattern of deductive reasoning is known as modus ponens
4 CHAPTER 1 logic x 5 2 6 Ï8 2 x 5 2 6 2Ï2 2 x 5 2s1 6 Ï2d 2 x 5 1 6 Ï2
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All
Some
Some
examPle 2
n Deductive reasoning and venn Diagrams
The validity of a deductive argument can be shown by use of a Venn diagram. A venn diagram is a diagram consisting of various overlapping figures contained within a rectangle (called U, the “universe”). To depict a statement of the form “All S are P” (or, equivalently, “If S, then P”), we draw two circles, one inside the other; the inner circle represents S (the “subject category”) and the outer circle represents P (the “predicate category”). This relationship is shown in Figure 1.1. Venn diagrams depicting “No S are P,” “Some S are P,” and “Some S are not P” are shown in Figures 1.2, 1.3, and 1.4, respectively.
The four basic sentences (or propositions) “All S are P,” “No S are P,” “Some S are P,” and “Some S are not P,” are known as the standard form categorical proposition types. A categorical proposition is a statement affirming or denying that one category, S, (the subject category) is wholly or partially contained in some other category, P (the predicate category). Each of the standard form categorical propositions has a title, as indicated in Figure 1.5.
anaLyzing a dedUctive argUment Construct a Venn diagram to verify the validity of the following argument:
1. All men are mortal.
2. Socrates is a man. Therefore, Socrates is mortal.
Premise 1 is of the form “All A are B” and can be represented by a diagram like that shown in Figure 1.6.
Premise 2 refers to a specific man, namely, Socrates. If we let x 5 Socrates, the statement “Socrates is a man” can then be represented by placing x within the circle labeled “men,” as shown in Figure 1.7. Because we placed x within the “men” circle, and all of the “men” circle is inside the “mortal” circle, the conclusion “Socrates is mortal” is inescapable; the argument is valid.
1.1 Deductive versus Inductive Reasoning 5
U S P
SP U
Figure 1.1 All S are P. (If S, then P.)
U S P
Figure 1.2 No S are P
U S P
Figure 1.3 Some S are P. (At least one S is P.)
Figure 1.4 Some S are not P. (At least one S is not P.)
SoluTIon
categorical Proposition type title venn diagram
S are P Universal affirmative Figure 1.1
S are P Universal negative Figure 1.2
No
S are P Particular affirmative Figure 1.3
S are not P Particular negative Figure 1.4
U men mortal Figure 1.6 All men are mortal. U x men mortal x 5 Socrates Figure 1.7 Socrates is mortal. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Figure 1.5 Standard form categorical proposition types.
Historical Note Aristotle 384–322 b
Aristotle was born in 384 b c in the small Macedonian town of Stagira, 200 miles north of Athens, on the shore of the Aegean Sea. Aristotle’s father was the personal physician of King Amyntas II, ruler of Macedonia. When he was seventeen, Aristotle enrolled at the Academy in Athens and became a student of the famed Plato.
Aristotle was one of Plato’s brightest students; he frequently questioned Plato’s teachings and openly disagreed with him. Whereas Plato emphasized the study of abstract ideas and mathematical truth, Aristotle was more interested in observing the “real world” around him. Plato often referred to Aristotle as “the brain” or “the mind of the school.” Plato commented, “Where others need the spur, Aristotle needs the rein.”
Aristotle stayed at the Academy for twenty years, until the death of Plato. Then the king of Macedonia invited Aristotle to supervise the education of his son Alexander, the future Alexander
examPle
3
x doctors men
x 5 My mother
Figure 1.8 My mother is a man.
. c .
the Great. Aristotle accepted the invitation and taught Alexander until he succeeded his father as ruler. At that time, Aristotle founded a school known as the Lyceum, or Peripatetic School. The school had a large library with many maps, as well as botanical gardens containing an extensive collection of plants and animals. Aristotle and his students would walk about the grounds of the Lyceum while discussing various subjects (peripatetic is from the Greek word meaning “to walk”).
Many consider Aristotle to be a founding father of the study of biology and of science in general; he observed and classified the behavior and anatomy of hundreds of living creatures. Alexander the Great, during his many military campaigns, had his troops gather specimens from distant places for Aristotle to study.
Aristotle was a prolific writer; some historians credit him with the writing
of over 1,000 books. Most of his works have been lost or destroyed, but scholars have re-created some of his more influential works, including Organon.
Aristotle’s collective works on syllogisms and deductive logic are known as Organon, meaning “instrument,” for logic is the instrument used in the acquisition of knowledge.
Collections of the New York Public Library
anaLyzing a dedUctive argUment Construct a Venn diagram to determine the validity of the following argument:
1. All doctors are men.
2. My mother is a doctor. Therefore, my mother is a man.
Premise 1 is of the form “All A are B”; the argument is depicted in Figure 1.8. No matter where x is placed within the “doctors” circle, the conclusion “My mother is a man” is inescapable; the argument is valid.
Saying that an argument is valid does not mean that the conclusion is true. The argument given in Example 3 is valid, but the conclusion is false. One’s mother cannot be a man! Validity and truth do not mean the same thing. An argument is valid if the conclusion is inescapable, given the premises. Nothing is said about the truth of the premises. Thus, when examining the validity of an argument, we are not determining whether the conclusion is true or false. Saying that an argument is valid merely means that, given the premises, the reasoning used to obtain the conclusion is logical. However, if the premises of a valid argument are true, then the conclusion will also be true.
6 CHAPTER 1 logic
SoluTIon
U
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examPle 5
anaLyzing a dedUctive argUment Construct a Venn diagram to determine the validity of the following argument:
1. All professional wrestlers are actors.
2. The Rock is an actor.
Therefore, The Rock is a professional wrestler.
Premise 1 is of the form “All A are B”; the “circle of professional wrestlers” is contained within the “circle of actors.” If we let x represent The Rock, premise 2 simply requires that we place x somewhere within the actor circle; x could be placed in either of the two locations shown in Figures 1.9 and 1.10.
SoluTIon
If x is placed as in Figure 1.9, the argument would appear to be valid; the figure supports the conclusion “The Rock is a professional wrestler.” However, the placement of x in Figure 1.10 does not support the conclusion; given the premises, we cannot logically deduce that “The Rock is a professional wrestler.” Since the conclusion is not inescapable, the argument is invalid.
Saying that an argument is invalid does not mean that the conclusion is false. Example 4 demonstrates that an invalid argument can have a true conclusion; even though The Rock (Dwayne Johnson) is a professional wrestler, the argument used to obtain the conclusion is invalid. In logic, validity and truth do not have the same meaning. Validity refers to the process of reasoning used to obtain a conclusion; truth refers to conformity with fact or experience.
Venn Diagrams and invalid Arguments
To show that an argument is invalid, you must construct a Venn diagram in which the premises are met yet the conclusion does not necessarily follow.
anaLyzing a dedUctive argUment Construct a Venn diagram to determine the validity of the following argument:
1. Some plants are poisonous.
2. Broccoli is a plant.
Therefore, broccoli is poisonous.
Premise 1 is of the form “Some A are B”; it can be represented by two overlapping circles (as in Figure 1.3). If we let x represent broccoli, premise 2 requires that we place x somewhere within the plant circle. If x is placed as in Figure 1.11, the argument would appear to be valid. However, if x is placed
1.1 Deductive versus Inductive Reasoning 7
SoluTIon
actors U x 5 The Rock professional wrestlers x Figure 1.9 actors U x x 5 The Rock professional wrestlers Figure 1.10
examPle 4
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Even though The Rock is a professional wrestler, the argument used to obtain the conclusion is invalid.
